(i) We work over the field of two elements. Define what is meant by a linear code of length $n$. What is meant by a generator matrix for a linear code?

Define what is meant by a parity check code of length $n$. Show that a code is linear if and only if it is a parity check code.

Give the original Hamming code in terms of parity checks and then find a generator matrix for it.

[You may use results from the theory of vector spaces provided that you quote them correctly.]

(ii) Suppose that $1 / 4>\delta>0$ and let $\alpha(n, n \delta)$ be the largest information rate of any binary error correcting code of length $n$ which can correct $[n \delta]$ errors.

Show that

$$1-H(2 \delta) \leqslant \liminf _{n \rightarrow \infty} \alpha(n, n \delta) \leqslant 1-H(\delta)$$

where

$$H(\eta)=-\eta \log _{2} \eta-(1-\eta) \log _{2}(1-\eta)$$

[You may assume any form of Stirling's theorem provided that you quote it correctly.]